## List of issues > Series «Mathematics». 2019. Vol. 29

##
The Completeness Criterion for Closure Operator
with the Equality Predicate Branching on the Set
of Multioperations on Two-Element Set

Multioperations are operations from a finite set A to set of all subsets of A. The usual composition operator leads to a continuum of closed sets. Therefore, the research of closure operators, which contain composition and other operations becomes necessary. In the paper, the closure of multioperations that can be obtained using the operations of adding dummy variables, identifying variables, composition operator, and operator with the equality predicate branching is studied. We obtain eleven precomplete closed classes of multioperations of rank 2 and prove the completeness criterion. The diagram of inclusions for one of the precomplete class is presented.

Vladimir Panteleev, Dr. Sci. (Phys.–Math.), Prof., Institute of mathematics, economics and informatics, Irkutsk State University, 664000, Irkutsk, K. Marks str., 1, tel.:+7(3952)242214, e-mail: vl.panteleyev@gmail.com

Leonid Riabets, Cand. Sci. (Phys.–Math.), Associate Professor, Institute of mathematics, economics and informatics, Irkutsk State University, 664000, Irkutsk, K. Marks str., 1, tel.:+7(3952)242214, e-mail: l.riabets@gmail.com

Panteleev V.I., Riabets L.V. The completeness criterion for closure operator with the equality predicate branching on the set of multioperations on two-element set. *The Bulletin of Irkutsk State University. Series Mathematics*, 2019, vol. 29, pp. 68-85. https://doi.org/10.26516/1997-7670.2019.29.68

- Doroslovacki R., Pantovic J., Vojvodic G. One Interval in the Lattice of Partial Hyperclones.
*Chechoslovak Mathematical Journal*, 2005, no. 55(3), pp. 719-724. https://doi.org/10.1007/s10587-005-0059-0. - Lau D.
*Function Algebras on Finite Sets. A basic course on many-valued logic and clone theory*. Berlin, Springer-Verlag, 2006, 668 p. - Lo Czu Kai. Maximal closed classes on the Set of Partial Many-valued Logic Functions.
*Kiberneticheskiy Sbornik*, Moscow, Mir Publ., 1988, vol. 25, pp. 131–141. (in Russian) - Lo Czu Kai. Completeness theory on Partial Many-valued Logic Functions.
*Kiberneticheskiy Sbornik*, Moscow, Mir Publ., 1988, vol. 25, pp. 142-157. (in Russian) - Machida H. Hyperclones on a Two-Element Set.
*Multiple-Valued Logic. An International Journal*, 2002, no. 8(4), pp. 495–501. https://doi.org/10.1080/10236620215294. - Machida H., Pantovic J. On Maximal Hyperclones on {0, 1} — a new approach.
*Proceedings of 38th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2008)*, 2008, pp. 32-37. - Marchenkov S.S. On the Expressibility of Functions of Many-Valued Logic in Some Logical-Functional Classes.
*Discrete Math. Appl*., 1999, vol. 9, no. 6, pp. 563-581. - Marchenkov S.S. Closure Operators with Predicate Branching.
*Bulletin of Moscow State University. Series 1. Mathematics and Mechanics*, 2003, no. 6, pp. 37–39. (in Russian) - Marchenkov S.S. The Closure Operator with the Equality Predicate Branching on the Set of Partial Boolean Functions.
*Discrete Math. Appl*., 2008, vol. 18, no. 4, pp. 381-389. https://doi.org/10.1515/DMA.2008.028. - Marchenkov S.S. The E-closure Operator on the Set of Partial Many-Valued Logic Functions.
*Mathematical problems in cybernetics*, Moscow, Fizmatlit, 2013, vol. 19, pp. 227–238. (in Russian) - Matveev S.S. Construction of All E-closed Classes of Partial Boolean Functions.
*Mathematical problems in cybernetics*, Moscow, Fizmatlit Publ., 2013, vol. 18, pp. 239-244. (in Russian) - Panteleev V.I. The Completeness Criterion for Depredating Boolean Functions.
*Vestnik of Samara State University. Natural Science Series*., 2009, no. 2 (68), pp. 60-79. (in Russian) - Panteleev V.I. Completeness Criterion for Underdetermined Partial Boolean Functions.
*Vestnik Novosibirsk State University. Series Mathematics*, 2009, vol. 9, no. 3, pp. 95-114. (in Russian) - Panteleev V.I., Riabets L.V. The Closure Operator with the Equality Predicate Branching on the Set of Hyperfunctions on Two-Element Set.
*The Bulletin of Irkutsk State University. Series Mathematics*, 2014, vol. 10, pp. 93-105. (in Russian) - Romov B.A. Hyperclones on a Finite Set.
*Multiple-Valued Logic. An International Journal*, 1998, vol. 3(2), pp. 285-300.